By reordering the rows and columns,
the matrix \(D\)  can be put into a “block diagonal form”,
where the blocks cannot be refined into a block diagonal sum
any further.
The \(p\) -blocks of \(G\)  correspond to the blocks in this
finest block diagonal form of \(D\) . To a \(p\) -block \(B\) 
associated to a block \(D_B\)  in the decomposition matrix \(D\) ,
we associate all the irreducible characters and simple \(FG\) -modules
labelling the rows and columns of \(D_B\) ; the
corresponding set of irreducible characters is denoted by \({\rm Irr}(B)\) .
There are also intrinsic definition of \(p\) -blocks as indecomposable ideals
in the group algebra, and a direct criterion for when
two irreducible characters belong to the same \(p\) -block (see for example
[1]}, [2]}, [3]}).
